### Table of Contents

## BCK-algebras

Abbreviation: **BCK**

### Definition

A \emph{BCK-algebra} is a structure $\mathbf{A}=\langle A,\cdot ,0\rangle$ of type $\langle 2,0\rangle$ such that

(1): $((x\cdot y)\cdot (x\cdot z))\cdot (z\cdot y) = 0$

(2): $x\cdot 0 = x$

(3): $0\cdot x = 0$

(4): $x\cdot y=y\cdot x= 0 \Longrightarrow x=y$

Remark: $x\le y \iff x\cdot y=0$ is a partial order, with $0$ as least element.

BCK-algebras provide algebraic semantics for BCK-logic, named after
the combinators B, C, and K by C. A. Meredith, see ^{1)}.

### Definition

A \emph{BCK-algebra} is a BCI-algebra $\mathbf{A}=\langle A,\cdot ,0\rangle$ such that

$x\cdot 0 = x$

##### Morphisms

Let $\mathbf{A}$ and $\mathbf{B}$ be BCK-algebras. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a homomorphism: $h(x\cdot y)=h(x)\cdot h(y)$ and $h(0)=0$

### Examples

Example 1:

### Basic results

### Properties

Classtype | quasivariety ^{2)} |
---|---|

Equational theory | |

Quasiequational theory | |

First-order theory | undecidable |

Locally finite | no |

Residual size | unbounded |

Congruence distributive | no |

Congruence modular | no |

Congruence n-permutable | no |

Congruence regular | no |

Congruence uniform | no |

Congruence extension property | no |

Definable principal congruences | no |

Equationally def. pr. cong. | no |

Amalgamation property | yes |

Strong amalgamation property | yes ^{3)} |

Epimorphisms are surjective |

### Finite members

$\begin{array}{lr}
f(1)= &1

f(2)= &1

f(3)= &3

f(4)= &14

f(5)= &88

f(6)= &775

\end{array}$

### Subclasses

### Superclasses

### References

^{1)}A. N. Prior, \emph{Formal logic}, Second edition, Clarendon Press, Oxford, 1962, p.316

^{2)}Andrzej Wronski,\emph{BCK-algebras do not form a variety}, Math. Japon., \textbf{28}, 1983, 211–213

^{3)}Andrzej Wronski,\emph{Interpolation and amalgamation properties of BCK-algebras}, Math. Japon., \textbf{29}, 1984, 115–121